1. Field of the Invention
The present invention relates generally to digital signal processing techniques, and particularly to an adaptive filter for system identification that uses a modified least mean squares algorithm with a variable step-size for fast convergence while preserving reasonable precision when the unknown system is sparse.
2. Description of the Related Art
In many electronic circuits, it is necessary to process an input signal through a filter to obtain the desired signal, e.g., to remove noise. The filter implements a transfer function. When the coefficients remain unchanged, the filter is static, and always processes the input signal in the same manner. However, in some applications it is desirable to dynamically change the transfer function in order to produce an output signal that is closer to the desired signal. This is accomplished by using an adaptive filter that compares the output signal of the filter to a reference signal that is an estimate of the desired signal, producing an error signal that is fed back and used to alter the coefficients of the transfer function. Adaptive filters are commonly implemented by digital signal processors, although they may also be implemented by an application specific integrated circuit (ASIC) or by field programmable gate arrays (FPGAs). Adaptive filters are used in a variety of applications, including channel estimation, noise cancellation, system identification, echo cancellation in audio and network systems, and signal prediction, to name a few.
In system identification applications, the goal is to specify the parameters of an unknown system model on the basis of measurements of the input signal and output response signals of the unknown system, which are used to generate error signals that adjust the adaptive filter to more closely correlate with the unknown system model. System identification refers to the process of learning more about the unknown system with each input-output cycle and adjusting the adaptive filter to incorporate the newly acquired knowledge about the unknown system.
Adaptive filters use a variety of algorithms in the feedback loop to recalculate and update the coefficients of the transfer function. One of the most commonly used algorithms is the least mean square (LMS) algorithm, which is easy to implement and generally produces a reasonably accurate output signal. The LMS algorithm, also known as the stochastic gradient algorithm, was introduced by Widrow and Huff around 1960, and basically provides that adjustments should be made at each sample time t according to:ht+1=ht+2 μetxt  (1)where h is a vector of the filter coefficients, e is the error signal, x is the input signal, and μ is a step size. The step size μ is usually chosen to be between 0 and 1. From equation (1), both the speed of convergence and the error in the adjustment are both proportional to μ. This results in a trade-off. The greater the value of μ, the faster the convergence, but the greater the adjustment error. On the other hand, the smaller the value of μ, the slower the convergence, but the closer the correlation to the unknown system. Both convergence time and the total misadjustment may also be affected by the length of the filter.
In system identification, the balance between convergence time and total misadjustment may also be affected by the quality of the unknown system. When the unknown system is sparse, many of the filter coefficients (tap weights controlling the magnitude each tap contributes to the output signal) will be zero or close to zero. Many attempts have been made to modify the LMS algorithm to exploit the sparseness of the system to produce faster convergence.
An algorithm recently proposed by Yilun Chen et al. in “Sparse LMS for System Identification”, Proc. of ICASSP '09 (2009), pp. 3125-3128, (the contents of which are hereby incorporated by reference in their entirety) referred to as the Reweighted Zero-Attracting LMS (RZA-LMS) algorithm, improves the performance of the LMS algorithm in system identification of sparse systems. However, the algorithm requires the use of a very small value of step size in order to converge. There continues to be a need for improved adjustment in the step size in algorithms used by adaptive filters for system identification.
Thus, an adaptive filter for system identification solving the aforementioned problems is desired.